Collatz Number System Question

Question

I was looking at the Collatz Conjecture and I thought of something:

If we denote two operators $a_n = 2n$ and $b_n = \frac{n-1}{3}$, then every number that converges using the Collatz Conjecture can be represented in the form of a continued subscript, like 5: $$b_{a_{a_{a_{a_{1}}}}}$$

And to make it easier to read, I'll remove the subscripts, making it:

$$baaaa1$$

If we remove the $1$, giving $baaaa$, and then replace $b$ as $0$ and $a$ as $1$, we get a psuedo-binary system. Thus $5$ is $01111$ and $7$ is $0101011011101111$. The longer it takes for a number to converge, the longer the number is written down.

I realize that this has an information density much much less than binary, but I was wondering: Given this number system, and given two numbers, $x$ and $y$, how can we determine what their sum, $x+y$ will be, or if it will even exist in this number system?

Answer 1

Adding the binary representations doesn't yield much in this case. But the representation you suggest e.g. baaaa1 is more interesting.

You could compress this system of representation and instead just record the number of a steps, as an index of sorts. Whenever there is a b step, you add another index. It would also be easier to run this in reverse (i.e. divide by two for a and 3n+1 for b).

A few examples:

5: {4} or baaaa

7: {1,2,2,3,4} or babaabaabaaaabaaaa

9: {2,1,2,2,3,4} or baababaabaaabaaaabaaaa

11: {2,2,3,4} or baabaabaaabaaaa

To clarify further, the representations tell you how to construct the number using the operations a and b starting from 1. i.e. 5 is (1 x 2 x 2 x 2 x 2 - 1) / 3. Each index in the "positional notation" is then a power of 2.

Some insight can be gained using the "positional" notation. For example if you investigate all numbers that only have one index, you'll see that only even indexes (powers) greater than 4 crop up. And that if you look at the binary representations of those numbers you'll see a pattern:

5: {4} or 101

21: {6} or 10101

85: {8} or 1010101

The patterns of allowable indexes and their binary representations are more complex in other cases where there are more steps, but they are finite. That means you could in theory construct the indeces in the positional notation directly from the binary representation of the number without going through the process of applying the Collatz operations. In the example above this obviously can be done.

Note that the "positional notation" is a truncation, because once you reach 1 you go into a loop of baabaabaa... So technically 5 would be {4,2,2,2,2...}.

The Collatz conjecture is then the same as saying that every starting number n has a unique (finite) representation in this system ending in 2,2,2...

Answer 2

After looking at it for a little bit, it does not make too much sense to "add" or "subtract" these numbers together in the way one normally adds and subtracts. As for checking what values are real or not, I have a few ideas.

I tested adding by using the numbers 5 (01111) and 4 (11). By adding normally, I received the answer (10010), which is a nonsense answer because you can not multiply an even number by 3 and add 1. I then tried simply moving the (11) in front of (01111). It made (1101111), which is 20. Doing the reverse (0111111) created 21. I then tried 'adding' 5 (01111) and 13 (011101111). Putting 5 in front of 13 created 69, however the reverse created a non-existent number because 80 does not produce a whole number after ${(n-1)}/{3}$.

Overall, concatenating numbers seems to work sometimes, although I am unsure if this counts as "addition" by any means.

To check if the numbers are real: if there are no easy consecutive zeros to rule out, for instance (11001111) is a fake number, then the best way I found to check it is through brute force. Follow the number from right to left and follow the process you mentioned earlier to find if it works or not.

There may be more interesting properties about these numbers or better ways to work with them. I would think it could be interesting to look further into this!